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In this work we present the generalization of some thermodynamic propertiesof the black body radiation (BBR) towards an $n-$dimensional Euclidean space.For this case the Planck function and the Stefan-Boltzmann law have alreadybeen given by Landsberg and de Vos and some adjustments by Menon and Agrawal.However, since then no much more has been done on this subject and we believethere are some relevant aspects yet to explore. In addition to the resultspreviously found we calculate the thermodynamic potentials, the efficiency ofthe Carnot engine, the law for adiabatic processes and the heat capacity atconstant volume. There is a region at which an interesting behavior of thethermodynamic potentials arise, maxima and minima appear for the $n-d$ BBRsystem at very high temperatures and low dimensionality, suggesting a possibleapplication to cosmology. Finally we propose that an optimality criterion in athermodynamic framework could have to do with the $3-d$ nature of the universe.

A possible cosmological application of some thermodynamic properties of the black body radiation in $n-$dimensional Euclidean spaces